
where we use \(d_i\)  to denote the GW data associated with the \(i\) -th observation, \(L(d_i|\theta )\)  is the likelihood of the data given the source parameters \(\theta \)  [1]}, [2]}, \(\pi _\mathrm {hier}(\theta |\Lambda )\)  is the population model defined in Sec. , and \(P_{\mathrm {det}}(\Lambda )\)  is the fraction of all astrophysical events which are observed and accounts for selection biases [3]}, [4]}.
The fraction \(P_{\mathrm {det}}(\Lambda )\)  scales as the surveyed space-time volume \(VT(\Lambda )\)  of the detector network for a binary black hole population with population hyperparameters \(\Lambda \) ;
we calculate \(VT(\Lambda )\)  analytically following [5]}, using a single-detector network with a median (over observing times from the first and second observing runs) LIGO Hanford noise curve and signal-to-noise ratio threshold of 8.
The overall likelihood in Eq. (REF ) can be broken into pieces associated with each generation,
\(\mathcal {L}_\mathrm {hier}(d_i|\Lambda ) = & {} \: \frac{1}{P_{\mathrm {det}}(\Lambda )} \left[\zeta _\textsf {1G+1G} {}(\Lambda )\mathcal {L}(d_i|\Lambda ,\textsf {1G+1G} {}) \right. \nonumber \\& +\left. {} \zeta _\textsf {1G+2G} {}(\Lambda )\mathcal {L}(d_i|\Lambda ,\textsf {1G+2G} {}) \right.\nonumber \\& +\left. {} \zeta _\textsf {2G+2G} {}(\Lambda )\mathcal {L}(d_i|\Lambda ,\textsf {2G+2G} {}) \right],\) 
